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Chapter 8

Principles Used in This Chapter1.Portfolio Returns and Portfolio Risk2.Systematic Risk and the Market Portfolio3.The Security Market Line and the CAPM

1. Calculate the expected rate of return and volatility for a portfolio of investments

2. Describe how diversification affects the returns to a portfolio of investments.

d d h f k f3. Understand the concept of systematic risk for an individual investment

4. Calculate portfolio systematic risk (beta).5. Estimate an investor’s required rate of return

using capital asset pricing model.

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Principle 2: There is a Risk-Return Tradeoff.◦ We extend our risk return analysis to consider

portfolios of risky investments and the beneficial effects of portfolio diversification on risk. ◦ In addition, we will learn more about what types ofIn addition, we will learn more about what types of

risk are associated with both higher and lower expected rates of return.

With appropriate diversification, we can lower the risk of the portfolio without lowering the portfolio’s expected rate of return.

k b l d b d f d Some risk can be eliminated by diversification, and those risks that can be eliminated are not rewarded in the financial marketplace.◦ The market will not reward you for bearing risk

needlessly

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To calculate a portfolio’s expected rate of return, we weight each individual investment’s expected rate of return using the fraction of the portfolio that is invested in each investmentinvestment.◦ Price per share×Number of shares/Value of Portfolio

Example 8.1 : If you invest ◦ 25%of your money in the stock of Citi bank (C) with

an expected rate of return of -32% ◦ 75% of your money in the stock of Apple (AAPL) with

an expected rate of return of 120%an expected rate of return of 120%◦ What will be the expected rate of return on this

portfolio?

Expected rate of return

= .25(-32%) + .75 (120%)

= 82%

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E(rportfolio) = the expected rate of return on a portfolio of n assetsportfolio of n assets.

Wi = the portfolio weight for asset i.◦ Percentage of the total portfolio by value

E(ri ) = the expected rate of return earned by asset i.

W1 × E(r1) = the contribution of asset 1 to the portfolio expected return.

Calculating a Portfolio’s Expected Rate of ReturnPenny Simpson has her first full-time job and is considering how to invest her savings. Her dad suggested she invest no more than 25% of her savings in the stock of her employer, Emerson Electric (EMR), so she is considering investing the remaining 75% in a combination of a risk-free investment in U.S. Treasury bills, currently paying 4%, and Starbucks (SBUX) common stock. Penny’s father has invested in the stock market for many years and suggested that Penny mighthas invested in the stock market for many years and suggested that Penny might expect to earn 9% on the Emerson shares and 12% from the Starbucks shares. Penny decides to put 25% in Emerson, 25% in Starbucks, and the remaining 50% in Treasury bills. Given Penny’s portfolio allocation, what rate of return should she expect to receive on her investment?

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Unlike expected return, standard deviation is not generally equal to the a weighted average of the standard deviations of the returns of investments held in the portfolio. Thi i b f di ifi ti ff t This is because of diversification effects.

The effect of reducing risks by including a large number of investments in a portfolio is called diversification.

f d f h As a consequence of diversification, the standard deviation of the returns of a portfolio is typically less than the average of the standard deviation of the returns of each of the individual investments.

The diversification gains achieved by adding more investments will depend on the degree of correlation among the investments.

h d f l d b The degree of correlation is measured by using the correlation coefficient.

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The correlation coefficient can range from -1.0 (perfect negative correlation), meaning two variables move in perfectly opposite directions to +1 0 (perfect positive correlation) which+1.0 (perfect positive correlation), which means the two assets move exactly together.

A correlation coefficient of 0 means that there is no relationship between the returns earned by the two assets.

As long as the investment returns are not perfectly positively correlated, there will be diversification benefits.

However, the diversification benefits will be t h th l ti lgreater when the correlations are low or

positive. The returns on most investment assets tend

to be positively correlated (tend to move up and down together).

1. A portfolio can be less risky than the average risk of its individual investments in the portfolio.

2. The key to reducing risk through diversification is b h dto combine investments whose returns do not

move together.

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Determine the expected return and standard deviation of the following portfolio consisting of two stocks that have a correlation coefficient of .75.

Portfolio Weight Expected Return

Standard Deviation

Apple .50 .14 .20

Coca-Cola .50 .14 .20

Expected Return = .5 (.14) + .5 (.14)

= .14 or 14%

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Standard deviation of portfolio = √ { (.52x.22)+(.52x.22)+(2x.5x.5x.75x.2x.2)}= √ .035= .187 or 18.7%

Correlation Coefficient

Evaluating a Portfolio’s Risk and ReturnSarah Marshall Tipton is considering her 401(k) retirement portfolio and wonders if she should move some of her money into international investments. To this point in her short working life (she graduated just four years ago), she has simply put her retirement savings into a mutual fund whose investment strategy mimicked the returns of the S&P 500 stock index (large company stocks). This fund has historically earned a return averaging 12% over the last 80 or so years, but recently the returns were depressed somewhat, as the economy was languishing in a mild recession. Sarah Is considering an international mutual fund that diversifies its holdings around the industrialized economies of the world and has averaged a 14% annual rate of return. The international fund’seconomies of the world and has averaged a 14% annual rate of return. The international fund s higher average return is offset by the fact that the standard deviation in its returns is 30% compared to only 20% for the domestic index fund. Upon closer investigation, Sarah learned that the domestic and international funds tend to earn high returns and low returns at about the same times in the business cycle such that the correlation coefficient is .75. If Sarah were to move half her money into the international fund and leave the remainder in the domestic fund, what would her expected portfolio return and standard deviation in portfolio return be for the combined portfolio?

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Evaluate the expected return and standard deviation of the portfolio where the correlation is assumed to be .20 and Sarah still places half of her money in each of the funds.

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We can visualize the expected return, standard deviation and weights as follows:

Investment Fund

Expected Return

Standard Deviation

Investment Weight

S&P500 12% 20% 50%

Sarah needs to determine the answers to place in the empty squares.

S&P500 fund

12% 20% 50%

International Fund

14% 30% 50%

Portfolio 100%

The portfolio expected return is a simple weighted average of the expected rates of return of the two investments given by equation 8-1.

The standard deviation of the portfolio can be calculated using equation 8-2. We are given the correlation to be equal to .20.

E(rportfolio) ( ) ( )= WS&P500 E(rS&P500) + WInternational E(rInternational)

= .5 (12) + .5(14)= 13%

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Standard deviation of Portfolio= √ { (.52x.22)+(.52x.32)+(2x.5x.5x.20x.2x.3)}√= √ {.0385}

= .1962 or 19.62%

The standard deviation of the portfolio is less than 25% at 19.62% because of the diversification benefits.

Since the correlation between the two funds is less than 1 combining the two funds into one portfoliothan 1, combining the two funds into one portfolio results in portfolio risk reduction.

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It would be an onerous task to calculate the correlations when we have thousands of possible investments.

l d l h Capital Asset Pricing Model or the CAPM provides a relatively simple measure of risk.

CAPM assumes that investors chose to hold the optimally diversified portfolio that includes all risky investments.

This optimally diversified portfolio that i l d ll f th ’ t iincludes all of the economy’s assets is referred to as the market portfolio.

According to the CAPM, the relevant risk of an investment relates to how the investment contributes to the risk of this market portfolio.

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To understand how an investment contributes to the risk of the portfolio, we categorize the risks of the individual investments into two categories:

S stematic risk and◦ Systematic risk, and◦ Unsystematic risk

The systematic risk component measures the contribution of the investment to the risk of the market. ◦ For example: War, hike in corporate tax rate.

The unsystematic risk is the element of risk that does not contribute to the risk of the market.

This component is diversified away when the investment is combined with other investments. ◦ For example: Product recall, labor strike, change of

management.

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Systematic risk is measured by the beta coefficient, ◦ Beta estimates the extent to which a particular

investment’s returns vary with the returns on the market portfolio.market portfolio.

In practice, it is estimated as the slope of a straight line (see figure 8-3)

Figure 8.3 cont.

Beta could be estimated using excel or financial calculator, or readily obtained from various sources on the internet (such as Yahoo Finance and Money Central.com)

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Table 8-1 illustrates the wide variation in Betas for various companies.

Utilities companies can be considered less risky because of their lower betas.

A 1% d i k ld l d 74% d i◦ A 1% drop in market could lead to a .74% drop in AEP and ◦ A much larger 2.9% drop in AAPL.

The portfolio beta measures the systematic risk of the portfolio

Portfolio beta is calculated by taking a simple weighted average of the betas for the i di id l i t t t i d i thindividual investments contained in the portfolio.◦ Same portfolio weights we have been using

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Example 8.2 Consider a portfolio that is comprised of four investments with betas equal to 1.5, .75, 1.8 and .60.

If you invest equal amount in each i t t h t ill b th b t f thinvestment, what will be the beta for the portfolio?

Portfolio Beta = .25(1.5) + .25(.75) + .25(1.8) + .25 (.6)= 1.16

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CAPM also describes how the betas relate to the expected rates of return that investors require on their investments.

h k h f h ll The key insight of CAPM is that investors will require a higher rate of return on investments with higher betas.

Figure 8-4 provides the expected returns and betas for a variety of portfolios comprised of

k f l d k f

])([)( fMportfoliofportfolio rrErrE

market portfolio and risk-free asset. However, the figure applies to all investments, not just portfolios consisting of the market and the risk-free rate.

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Figure 8.4 cont.

The straight line relationship between the betas and expected returns in Figure 8-4 is called the security market line (SML),◦ Its slope is often referred to as the reward to risk ratio.

SML is a graphical representation of the CAPM.

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SML can be expressed as the following equation, which is also referred to as the CAPM pricing equation:

The higher the systematic risk of an investment, the higher the expected rate of return an investor would require to invest in the asset.

This is consistent with Principle 2: There is a Risk-Return Tradeoff.

Example 8.2 What will be the expected rate of return on AAPL stock with ◦ a beta of 1.49 ◦ if the risk-free rate of interest is 2% and ◦ if the market risk premium is estimated to be 8%?◦ if the market risk premium is estimated to be 8%?

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E(rAAPL ) = .02 + 1.49 (.08)= .1392 or 13.92%

Estimating the Expected Rate of Return Using the CAPMJerry Allen graduated from the University of Texas with a finance degree in the spring of 2010 and took a job with a Houstonbased investment banking firm as a financial analyst. One of his first assignments is to investigate the investor-expected rates of return for three technology firms: Apple (APPL), Dell (DELL), and Hewlett Packard (HPQ). Jerry’s supervisor suggests that he make his estimates using the CAPM where the risk-free rate is 4 5% the expected returnestimates using the CAPM where the risk free rate is 4.5%, the expected return on the market is 10.5%, and the risk premium for the market as a whole (the difference between the expected return on the market and the risk-free rate) is 6%. Use the two estimates of beta provided for these firms in Table 8.1 to calculate two estimates of the investor-expected rates of return for the sample firms.

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Estimate the expected rates of return for the three utility companies, found in Table 8-1, using the 4.5% risk-free rate and market risk premium of 6%

10.0%

12.0%

14.0%

16.0%

18.0%

d R

etu

rn

0.0%

2.0%

4.0%

6.0%

8.0%

0 0.5 1 1.5 2 2.5

Exp

ecte

BETA

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The graph shows that as beta increases, the expected return also increases.

When beta = 0, the expected return is equal h k f fto the risk free rate of 4.5%.

We can determine the required rate of return by using CAPM equation 8-6. The betas for the three utilities companies (Yahoo Finance estimates) are:

◦ AEP = 0.74◦ DUK = 0.40◦ CNP = 0.82

E(return): (AEP) = 4.5% + 0.74(6) = 8.94% E(return): (DUK) = 4.5% + 0.40(6) = 6.9% E(return): (CNP) = 4.5% + 0.82(6) = 9.42%

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The expected rates of return on the stocks vary depending on their beta.

Higher the beta, higher is the expected return.